Calculus

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I recommend this book: A Course of Modern Analysis by Whittaker and Watson. You may also find this book at Google Books. This book is a hundred years old and is considered the classic calculus book.


Derivatives

Definition of Derivative

f'(x)=\lim _{{\Delta x\to 0}}{\frac  {f(x+\Delta x)-f(x)}{\Delta x}}, provided the limit exists.

For the following problems, find the derivative using the definition of the derivative.

solution f(x)=x\,

solution f(x)=2x^{3}\,

solution f(x)={\sqrt  {x}}\,

solution f(x)={\frac  {1}{x}}\,

solution f(x)=\sin x\,

solution h(x)=f(x)+g(x)\,

Power Rule

{\frac  {d}{dx}}(x^{n})=nx^{{n-1}}\,

For the following problems, compute the derivative of y=f(x)\, with respect to x\,

solution f(x)=150\pi \,

solution f(x)=7x^{4}\,

solution f(x)=27x^{2}+{\frac  {12}{\pi }}x+e^{{e^{{\pi }}}}\,

solution f(x)=8x^{5}+4x^{4}+6x^{3}+x+7\,

solution f(x)={\frac  {1}{{\sqrt[ {4}]{x}}}}\,

solution f(x)=7x^{{-4}}+12x^{{-2}}-x^{{{\frac  {1}{2}}}}+6x+\pi x^{{{\frac  {3}{5}}}}+x^{{-{\frac  {3}{4}}}}\,

solution Derive the power rule for positive integer powers from the definition of the derivative (Hint: Use the Binomial Expansion)

Product Rule

{\frac  {d}{dx}}(ab)=ab'+a'b\,

For the following problems, compute the derivative of y=f(x)\, with respect to x\,

solution f(x)=x\sin x\,

solution f(x)=x^{3}\cos x\,

solution f(x)=\tan x\sec x\,

solution f(x)=2x\sin(2x)\,

solution f(x)=(x)(x+7)(x-12)\,

solution Derive the Product Rule using the definition of the derivative

Quotient Rule

\left({\frac  {a}{b}}\right)'={\frac  {ba'-ab'}{b^{2}}}\,

For the following problems, compute the derivative of y=f(x)\, with respect to x

solution f(x)={\frac  {x}{x+1}}\,

solution f(x)={\frac  {x^{2}}{\sin x}}\,

solution f(x)={\frac  {\sin ^{2}x}{x^{3}}}\,

solution f(x)={\frac  {x\sin x}{e^{x}}}\,

solution f(x)={\frac  {x+7}{(x-6)(x+2)}}\,

solution Derive the Quotient Rule formula. (Hint: Use the Product Rule).

Generalized Power Rule

{\frac  {d}{dx}}(f(x))^{n}=n(f(x))^{{n-1}}f'(x)\,

For the following problems, compute the derivative of y=f(x)\, with respect to x

solution f(x)=(x^{3}+7x)^{3}\,

solution f(x)=\sin ^{4}(x)\,

solution f(x)=\left(\ln x\right)^{{-4}}\,

solution f(x)=\tan ^{2}x+8(x^{2}+4x+3)^{9}+\sec ^{3}x\,

solution f(x)=\left({\frac  {5x}{7x+9}}\right)^{3}\,


Chain Rule

{\frac  {d}{dx}}f(g(x))=f'(g(x))g'(x)\,

For the following problems, compute the derivative of y=f(x)\, with respect to x

solution f(x)=\ln(7x^{2}e^{x}\sin x)\,

solution f(x)=\sin ^{2}(7x+5)+\cos ^{2}(7x+5)\,

solution f(x)=6e^{{3x}}\tan(5x)\,

solution f(x)=\ln(\sin(e^{x}))\,

solution f(x)=\sin(\cos(\tan x))\,

solution f(x)=e^{{x^{2}}}\sin(14x)-\cos(e^{x})\,

solution f(x)={\frac  {{\frac  {\sin(5x)}{(x^{2}+1)^{2}}}}{\cos ^{3}(3x)-1}}\,

solution \left(f(g(x))\right)'=f'(g(x))g'(x)

solution {\sqrt[ {}]{x+{\sqrt[ {}]{x+{\sqrt[ {}]{x}}}}}}

solution f(x)=x^{2}{\sqrt  {9-x^{2}}}\,

Implicit Differentiation

For the following problems, compute the derivative of y=f(x)\, with respect to x

solution \sin(xy)=x\,

solution x+xy+x^{2}+xy^{2}=0\,

solution y=x(y+1)\,

solution yx=x^{y}\, where y\, is a function of x\,


Logarithmic Differentiation

For the following problems, compute the derivative of y=f(x)\, with respect to x

solution f(x)=4^{{\sin x}}\,

solution f(x)=x^{x}\,

solution f(x)=x^{{x^{{{\cdot }^{{{\cdot }^{{\cdot }}}}}}}}\,

solution f(x)=g(x)^{{h(x)}}\, for any functions g(x)\, and h(x)\, where g(x)\neq 0\,

Second Fundamental Theorem of Calculus

{\frac  {d}{dx}}\int _{{a}}^{{x}}f(t)\,dt=f(x)\,

For the following problems, compute the derivative of y=f(x)\, with respect to x

solution f(x)=\int _{{0}}^{{x}}e^{t}\,dt\,

solution f(x)=\int _{{5}}^{{x}}27t^{t}\sin(t-1)\,dt\,

solution f(x)=\int _{{4}}^{{x^{2}}}\sin(e^{t})\,dt\,

solution f(x)=\int _{{x^{2}}}^{{3x^{4}}}\cos(t)\,dt\,

solution Give a proof of the theorem

Applications of Derivatives

Slope of the Tangent Line

solution Find the slope of the tangent line to the graph f(x)=6x when x=3.

solution Find the slope of the tangent line to the graph f(x)=\sin ^{2}x when x=\pi .

solution Find the slope of the tangent line to the graph f(x)=\cos x+x^{5} when x={\frac  {\pi }{2}}.

solution Find the equation of the tangent line to the graph f(x)=xe^{x}+x+5 when x=0.

solution Find the slope of the tangent line to the graph f(x)=x^{2} when x=0,1,2,3,4,5,6.

solution Find the slope of the tangent line to the graph x^{2}+y^{2}=9 at the point (0,-3).

solution Find the equation of the tangent line to the graph xy=y^{2}x^{2}+y+x+2 at the point (0,-2).


Extrema (Maxima and Minima)

solution Find the absolute minimum and maximum on [-1,5] of the function f(x)=(1-x)e^{x}.

solution Find the absolute minimum and maximum on \left[-{\frac  {\pi }{2}},{\frac  {\pi }{2}}\right] of the function f(x)=\sin(x^{2}).

solution Find all local minima and maxima of the function f(x)=x^{2}.

solution Find all local minima and maxima of the function f(x)={\frac  {x^{2}-1}{x}}.

solution If a farmer wants to put up a fence along a river, so that only 3 sides need to be fenced in, what is the largest area he can fence with 100 feet of fence?

solution If a fence is to be made with three pens, the three connected side-by-side, find the dimensions which give the largest total area if 200 feet of fence are to be used.

solution Find the local minima and maxima of the function f(x)={\sqrt  {x}}.

Related Rates

solution A clock face has a 12 inch diameter, a 5.5-inch second hand, a 5 inch minute hand and a 3 inch hour hand. When it is exactly 3:30, calculate the rate at which the distance between the tip of any one of these hands and the 9 o'clock position is changing.

solution A spherical container of r meters is being filled with a liquid at a rate of \rho \,{{\rm {m}}}^{3}/{{\rm {min}}}. At what rate is the height of the liquid in the container changing with respect to time?


Projectile Motion

Integrals

Riemann Sums

Integration by Substitution

solution \int {\frac  {2x}{x^{2}+1}}\,dx\,

solution \int x^{2}\sin x^{3}\,dx\,

solution \int \cot x\,dx\,

solution \int \tan x\sec ^{2}x\,dx\,

solution \int {\frac  {\ln x^{2}}{x}}\,dx\,

solution \int _{{0}}^{{3}}{\frac  {2x+1}{x^{2}+x+7}}\,dx\,

solution \int _{{1}}^{{e^{{\pi }}}}{\frac  {\sin(\ln x)}{x}}\,dx\,

solution \int {\frac  {x}{{\sqrt  {4+x^{2}}}}}\,dx\,

solution \int {\frac  {x}{1-x^{2}}}\,dx\,


Integration by Parts

\int u\,dv=uv-\int v\,du\,

solution \int \ln x\,dx\,

solution \int x\sin(x)\,dx\,

solution \int \arctan(2x)\,dx\,

solution \int e^{x}\sin x\,dx\,

solution \int x^{2}e^{x}\,dx\,

solution \int (x^{3}+1)\cos x\,dx\,

solution \int x^{4}\sin x\,dx\,

solution \int {\frac  {\ln x}{x}}\,dx\,

Trigonometric Integrals

solution \int \sin ^{2}(x)\,dx\,

solution \int \tan ^{2}(x)\,dx\,

solution \int \sin ^{5}x\cos ^{5}x\,dx\,

solution \int \sin ^{2}x\cos ^{3}x\,dx\,

solution \int \sin ^{2}x\cos ^{2}x\,dx\,

solution \int \tan ^{2}x\sec ^{4}x\,dx\,

solution \int \tan ^{3}x\sec ^{3}x\,dx\,


Trigonometric Substitution

solution \int {\frac  {x\,dx}{{\sqrt  {3-2x-x^{2}}}}}

solution \int \operatorname{arcsec} x\,dx\,

solution \int {\frac  {1}{{\sqrt  {4x-x^{2}}}}}\,dx\,

solution \int x\arcsin x\,dx\,

solution \int {\frac  {x}{1-x^{2}}}\,dx\,

solution \int {\frac  {1}{(x^{2}+1)^{{{\frac  {3}{2}}}}}}\,dx\,

solution \int {\frac  {{\sqrt  {x^{2}-1}}}{x}}\,dx\,

solution \int {\frac  {x}{{\sqrt  {4+x^{2}}}}}\,dx\,


Partial Fractions

solution \int {\frac  {1}{x^{2}-1}}\,dx\,

solution \int {\frac  {1}{(x+1)(x+2)(x+3)}}\,dx\,

solution \int {\frac  {x!}{(x+n)!}}\,dx\,

solution \int {\frac  {x}{1-x^{2}}}\,dx\,

solution \int {\frac  {dx}{\sin ^{2}(x)-\cos ^{2}(x)}}

Special Functions

solution A ball is thrown up into the air from the ground. How high will it go?

solution Let f\, be a continuous function for x\geq a\,.
Show that \int _{a}^{x}\int _{a}^{s}f(y)\,dy\,ds=\int _{a}^{x}f(y)(x-y)\,dy\,

solution Evaluate \int e^{{x^{2}}}\,dx\,

solution \int _{0}^{\infty }3^{{-4z^{2}}}dz\,

solution \int _{0}^{\infty }x^{m}e^{{-ax^{n}}}dx\,

solution \int _{0}^{\infty }x^{4}e^{{-x^{3}}}\,dx\,

solution \int _{0}^{\infty }e^{{-pt}}{\sqrt  {t}}\,dt\,

Applications of Integration

Area Under the Curve

solution Find the area under the curve f(x)=x^{2} on the interval [-1,1].

solution Find the total area between the curve f(x)=\cos x and the x-axis on the interval [0,2\pi ].

solution Find the area under the curve f(x)=x^{3}+4x^{2}-7x+8 on the interval [0,1].

solution Using calculus, find the formula for the area of a rectangle.

solution Derive the formula for the area of a circle with arbitrary radius r.

Volume

Disc Method

The disc method is a special case of the method of cross-sectional areas to find volumes, using a circle as the cross-section.

To find the volume of a solid of revolution, using the disc method, use one of the two formulas below. R(x) is the radius of the cross-sectional circle at any point.

If you have a horizontal axis of revolution

V=\pi \int _{{a}}^{{b}}[R(x)]^{2}\,dx

If you have a vertical axis of revolution

V=\pi \int _{{c}}^{{d}}[R(y)]^{2}\,dy

solution Find the volume of the solid generated by revolving the line y=x around the x-axis, where 0\leq x\leq 4.

solution Find the volume of the solid generated by revolving the region bounded by y=x^{2} and y=4x-x^{2} around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by y=x^{2} and y=x^{3} around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by y=x^{2} and y=x^{3} around the y-axis.

solution Find the volume of the solid generated by revolving the region bounded by y={\sqrt  {x}}, the x-axis and the line x=4 around the x-axis.

Shell Method

Cross-sectional Areas

solution Find the volume, on the interval 0\leq x\leq 3, of a 3-D object whose cross-section at any given point is a square with side length x^{2}-9x.

solution Find the volume, on the interval 0<x<2\pi , of a 3-D object whose cross-section at any given point is an equilateral triangle with side length \sin {\frac  {x}{2}}.

solution Find the volume of a cylinder with radius 3 and height 10.

Arc Length

L=\int _{a}^{b}{\sqrt  {1+\left({\frac  {dy}{dx}}\right)^{2}}}\,dx=\int _{p}^{q}{\sqrt  {\left({\frac  {dx}{dt}}\right)^{2}+\left({\frac  {dy}{dt}}\right)^{2}}}\,dt


solution Calculate the arc length of the curve y=x^{2} from x=0 to x=4.

solution Determine the arc length of the curve given by x = t cos t , y = t sin t from t=0

solution Calculate the arc length of y=cosh x from x=0 to x

Mean Value Theorem

f'(c)={\frac  {f(b)-f(a)}{b-a}}.

solution Find the average value of the function f(x)=e^{{2x}} on the interval [0,4].

solution Find the average value of the function 2sec^{2}x on the interval [0,{\frac  {\pi }{4}}].

solution Find the average speed of a car, starting at time 0, if it drives for 5 hours and its speed at time t (in hours) is given by s(t)=5t^{2}+7t+e^{t}.

solution Find the average value of the function \sin ^{6}tcos^{3}t on the interval [0,{\frac  {\pi }{2}}].

solution Deduce the Mean Value Theorem from Rolle's Theorem.

Series of Real Numbers

Sequences

nth Term Test

If the series \sum _{{n=1}}^{{\infty }}a_{n} converges, then \lim _{{n\rightarrow \infty }}a_{n}=0.

Note: This leads to a test for divergence for those series whose terms do not go to 0 but it does not tell us if any series converges.

solution Discuss the convergence or divergence of the series with terms \{-1,1,-1,1,-1,1,...\}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n}} and \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{2}}}.

solution Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}\sin n.

Telescopic Series

If \{a_{k}\} is a convergent real sequence, then \sum _{{k=1}}^{{\infty }}(a_{k}-a_{{k+1}})=a_{1}-\lim _{{k\rightarrow \infty }}a_{k}.

solution Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}\left({\frac  {1}{k+7}}-{\frac  {1}{k+8}}\right).

solution Discuss the convergence or divergence of the series \sum _{{k=0}}^{{\infty }}{\frac  {2}{(k+1)(k+3)}}.

solution Discuss the convergence or divergence of the series \sum _{{k=2}}^{{\infty }}\ln \left({\frac  {k(k+2)}{(k+1)^{2}}}\right).

solution Discuss the convergence or divergence of the series \sum _{{k=4}}^{{\infty }}\left({\frac  {1}{k}}-{\frac  {1}{k+2}}\right).

solution Discuss the convergence or divergence of the series \sum _{{k=0}}^{{\infty }}\left(e^{{-k}}-e^{{-(k+1)}}\right).

solution Discuss the convergence or divergence of the series \sum _{{k=4}}^{{\infty }}\left(e^{{-k+3}}-e^{{-k+1}}\right).

Geometric Series

The series \sum _{{n=0}}^{{\infty }}r^{n} converges if -1<r<1 and, moreover, it converges to {\frac  {1}{1-r}}. For any other value of r, the series diverges. More generally, the finite series, \sum _{{n=a}}^{{b}}r^{n}={\frac  {r^{a}-r^{{b+1}}}{1-r}} for any value of r.

solution Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}\left({\frac  {1}{4}}\right)^{n}.

solution Discuss the convergence or divergence of the series \sum _{{n=3}}^{{\infty }}\left({\frac  {2}{3}}\right)^{n}.

solution Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}1.5^{n}.

solution Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}\left({\frac  {1}{6}}\right)^{{n+2}}.

solution Discuss the convergence or divergence of the series \sum _{{n=10}}^{{\infty }}\left({\frac  {3}{5}}\right)^{n}.

solution Discuss the convergence or divergence of the series \sum _{{n=7}}^{{\infty }}\left(-{\frac  {1}{2}}\right)^{n}.

solution Discuss the convergence or divergence of the series \sum _{{n=7}}^{{\infty }}\left[\left(-{\frac  {4}{7}}\right)^{{n+3}}+\left({\frac  {1}{3}}\right)^{{n-2}}\right]\,.

solution Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}{\frac  {3^{{n+1}}}{7^{n}}}.

solution Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}\left({\frac  {1}{3}}\right)^{{2n}}.

Integral Test

If the function f is positive, continuous, and decreasing for x\geq 1, then

\sum _{{n=1}}^{{\infty }}f(n)\, and \int _{{1}}^{{\infty }}f(x)\,dx\,

converge together or diverge together.

Notice that if f were negative, continuous, and increasing this is also true since such a function would simply be the negative of some function which is positive, continuous, and decreasing and multiplying by -1 will not change the convergence of a series.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {1}{n+1}}.

solution Discuss the convergence of \sum _{{n=0}}^{{\infty }}e^{{-3n}}.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{5}}}.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {n}{e^{n}}}.

solution Explain why the integral test is or is not applicable to \sum _{{n=1}}^{{\infty }}n^{2}.

solution Explain why the integral test is or is not applicable to \sum _{{n=1}}^{{\infty }}-{\frac  {1}{n^{2}}}.

solution Explain why the integral test is or is not applicable to \sum _{{n=1}}^{{\infty }}{\frac  {|\sin n|+1}{\ln(n+1)}}.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{p}}}.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{{{\frac  {7}{3}}}}}}.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {1}{{\sqrt[ {3}]{n}}}}.

Comparison of Series

Direct Comparison
If 0<a_{n}\leq b_{n} for all n
If \sum _{{n=1}}^{{\infty }}b_{n} converges, then \sum _{{n=1}}^{{\infty }}a_{n} also converges.
2. If \sum _{{n=1}}^{{\infty }}a_{n} diverges, then \sum _{{n=1}}^{{\infty }}b_{n} also diverges.
Limit Comparison
Suppose a_{n}>0, b_{n}>0 and \lim _{{n\rightarrow \infty }}{\frac  {a_{n}}{b_{n}}}=L where L is finite and positive.
Then \sum _{{n=1}}^{{\infty }}a_{n} and \sum _{{n=1}}^{{\infty }}b_{n} either both converge or both diverge.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}+4}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {\ln n}{n-3}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {2n^{2}}{4n^{4}+8n^{3}+3}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n{\sqrt  {n^{2}+1}}}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n+4}{(n+2)(n+1)}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\sin {\frac  {1}{n}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n+1}{n\times 3^{{n+1}}}}.

Dirichlet's Test

Let a_{k},b_{k}\in {\mathbb  {R}} for k\in {\mathbb  {N}}.

If the sequence of partial sums s_{n}=\sum _{{k=1}}^{{n}}a_{k} is bounded and b_{k}\downarrow 0 as k\rightarrow \infty , then \sum _{{k=1}}^{{\infty }}a_{k}b_{k} converges.

solution Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {\sin {\frac  {k\pi }{2}}}{k}}.

solution Discuss the convergence or divergence of the series \sum _{{k=2}}^{{\infty }}{\frac  {\sin {\frac  {k\pi }{3}}}{\ln k}}.

solution Discuss the convergence or divergence of the series \sum _{{k=7}}^{{\infty }}\left(-{\frac  {1}{2}}\right)^{k}.

solution Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{k2^{k}}}.

solution Discuss the convergence or divergence of the series \sum _{{k=3}}^{{\infty }}{\frac  {1}{\ln(\ln k)}}\cos \left({\frac  {k\pi }{3}}\right).

solution Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}a_{k}{\frac  {1}{k^{3}}} where a_{k}=\{7,4,6,3,-10,-10,7,4,6,3,-10,-10,...\}.

solution Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}\sin {\frac  {2k\pi }{n}}{\frac  {1}{\ln(\ln(\ln k))}} for any integer n\geq 1.

solution Discuss the convergence or divergence of the series \sum _{{k=3}}^{{\infty }}{\frac  {\ln(\ln k)}{\ln k}}a_{k} where \{a_{k}\}=\{1,-1,1,2,-3,1,2,4,-7,1,-1,1,2,-3,1,2,4,-7,...\}.

Alternating Series

If a_{n}, then the alternating series

\sum _{{n=1}}^{{\infty }}(-1)^{n}a_{n}\, and \sum _{{n=1}}^{{\infty }}(-1)^{{n+1}}a_{n}\,

converge if the absolute value of the terms decreases and goes to 0.

solution Discuss the convergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {(-1)^{n}}{n}}\,.

solution Discuss the convergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {(-1)^{{n+1}}n^{2}}{n^{2}+1}}\,.

solution Discuss the convergence of the series \sum _{{n=1}}^{{\infty }}\cos(n\pi )\,.

solution Discuss the convergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {(-1)^{n}}{{\sqrt  {n}}}}\,.

solution Discuss the convergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {(-1)^{{n+1}}n^{2}}{e^{n}}}\,.

solution Discuss the convergence of \sum _{{n=2}}^{{\infty }}{\frac  {(-1)^{n}}{\ln n}} including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {(-1)^{{n+1}}}{n^{2}+3n+4}} including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of \sum _{{n=1}}^{{\infty }}{\frac  {(-1)^{n}}{n!}} including whether the sum converges absolutely or conditionally.

Ratio Test

For the infinite series \sum _{{n=1}}^{{\infty }}a_{n},

1. If 0\leq \lim _{{n\rightarrow \infty }}\left|{\frac  {a_{{n+1}}}{a_{n}}}\right|<1\,, then the series converges absolutely.

2. If \lim _{{n\rightarrow \infty }}\left|{\frac  {a_{{n+1}}}{a_{n}}}\right|>1\,, then the series diverges.

3. If \lim _{{n\rightarrow \infty }}\left|{\frac  {a_{{n+1}}}{a_{n}}}\right|=1\,, then the test is inconclusive.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {2^{n}}{(2n)!}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n}{3^{{n+1}}}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {3^{n}}{n^{2}+2}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n}{(n+2)!}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n4^{n}}{n!}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n!}{4^{n}}}.

solution Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n^{k}a^{n}}{n!}}.

Root Test

Let a_{k}\in {\mathbb  {R}} and r=\limsup _{{k\rightarrow \infty }}|a_{k}|^{{\frac  {1}{k}}}

1. If r<1\,, then \sum _{{k=1}}^{{\infty }}a_{k} converges absolutely.

2. If r>1\,, then \sum _{{k=1}}^{{\infty }}a_{k} diverges.

3. If r=1\,, this test is inconclusive.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\left({\frac  {n}{2n+1}}\right)^{n}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}e^{{-n}}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n}{3^{n}}}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\left({\frac  {2n}{100}}\right)^{n}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {(n!)^{n}}{(n^{n})^{2}}}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\left({\frac  {7n}{12n-6}}\right)^{{2n}}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\left({\frac  {15n-6}{4n+2}}\right)^{{7n}}.

solutionDiscuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n!}{(n+1)^{n}}}\left({\frac  {23}{50}}\right)^{n}.

Cauchy Condensation Test

The series \sum _{{k=1}}^{{\infty }}a_{k} and \sum _{{k=1}}^{{\infty }}2^{k}a_{{2^{k}}} converge or diverge together.

solutionDiscuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{k}}.

solutionDiscuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{\ln k}}.

solutionDiscuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{k^{2}}}.

solutionDiscuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{{\sqrt  {2^{k}}}}}.

solutionDiscuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{(2^{k})^{n}}}, where n is some real number.

Logarithmic Test

Raabe's Test

Series of Real Functions

solution Find the infinite series expansion of {\frac  {1}{(1+x)^{a}}}\,

solution Investigate the convergence of this series: \sum _{{k=1}}^{\infty }{\frac  {1}{k(k+1)}}\,

solution Investigate the convergence of this series: 3-2+{\frac  {4}{3}}-{\frac  {8}{9}}+...+3\left(-{\frac  {2}{3}}\right)^{k}+...\,

solution Find the upper limit of the sequence \left\{x_{n}\right\}_{{n=1}}^{\infty },x_{n}=(-1)^{n}n\,

solution Find the upper limit of the sequence \left\{x_{n}\right\}_{{n=1}}^{\infty },x_{n}=(-1)^{n}\left({\frac  {2n}{n+1}}\right)\,

solution Evaluate \sum _{{k=0}}^{n}{n \choose k}^{2}\,

solution Evaluate \sum _{{m=1}}^{\infty }\sum _{{n=1}}^{\infty }{\frac  {m^{2}\,n}{3^{m}\left(m\,3^{n}+n\,3^{m}\right)}}.

solution Find the upper limit of the sequence \left\{x_{n}\right\}_{{n=1}}^{\infty },x_{n}={\frac  {1}{n^{2}}}\,

solution Find the upper limit of the sequence \left\{x_{n}\right\}_{{n=1}}^{\infty },x_{n}=n\sin \left({\frac  {n\pi }{2}}\right)\,

solution Evaluate \sum _{{n=0}}^{\infty }\left({\frac  {i}{3}}\right)^{n}\,

solution Determine the interval of convergence for the power series \sum _{{n=0}}^{\infty }{\frac  {n(3x-4)^{n}}{{\sqrt[ {3}]{n^{4}}}(2x)^{{n-1}}}}.


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